More Integrate woes
- To: mathgroup at smc.vnet.net
- Subject: [mg9584] More Integrate woes
- From: NOHAMcrose at c2.telstra-mm.net.au (Colin Rose)
- Date: Thu, 13 Nov 1997 01:40:48 -0500
- Organization: Theoretical Research Institute
- Sender: owner-wri-mathgroup at wolfram.com
In any statistical setting, calculation of the Gaussian distribution function is extremely important. Under Mathematica v3, this is something of a disaster. To summarise the problem: Under v3: ________ Expressions such as: aa = Integrate[Exp[-x^2],{x,-Infinity,y}, GenerateConditions->False] return output *of form*: 1 - Erf[Sqrt[y^2]] Our user now seeks numerical output. S/he enters: (aa /. y -> 3) == (aa/. y -> -3) True This is clearly FALSE. BY CONTRAST, under v2.2: _______________________ aa = Integrate[Exp[-x^2],{x,-Infinity,y}] returned output of form: 1 + Erf[y] which is correct for all real y. No problem. ________ The problem with v3 here is not only that setting GenerateConditions->False yields incorrect results for one of the most important and common integration problems, but that even if we set GenerateConditions->True the resulting conditional statement treats the distribution as if it is asymmetrical, when the normal distribution itself is clearly symmetrical. For instance: aa = Integrate[Exp[-x^2],{x,-Infinity,y}, GenerateConditions->True] returns If[ y < 0, -(1/2)*Sqrt[Pi]*(-1 + Erf[Sqrt[y^2]]) , Integrate[E^(-x^2), {x, -Infinity, y}] ] Choosing y < 0 as opposed to y > 0 as a basis for conditional output is completely arbitrary, and precisely the cause of the GenerateConditions->False bug. It would be great to hear that this will be fixed in v3.1. [ Co-author Murray Smith discovered these oddities and has been cursing wolfies ever since. ] Cheerio Colin -- Colin Rose tr(I) - Theoretical Research Institute ______________________________________ NOHAMcrose at c2.telstra-mm.net.au http://www.usyd.edu.au/su/tri/